hyperbolic cases
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11: 36.4 Bifurcation Sets
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Special Cases
… ►Hyperbolic umbilic bifurcation set (codimension three): … ►Hyperbolic umbilic cusp line (rib): … ►§36.4(ii) Visualizations
… ►12: 36.2 Catastrophes and Canonical Integrals
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Normal Forms for Umbilic Catastrophes with Codimension
… ►(hyperbolic umbilic). ►Canonical Integrals
… ►§36.2(ii) Special Cases
… ►§36.2(iv) Addendum to 36.2(ii) Special Cases
…13: 18.19 Hahn Class: Definitions
14: 13.8 Asymptotic Approximations for Large Parameters
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►Special cases are
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►For more asymptotic expansions for the cases
see Temme (2015, §§10.4 and 22.5)
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►where , and .
…For the case
the transformation (13.2.40) can be used.
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►uniformly with respect to bounded positive values of in each case.
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15: 19.10 Relations to Other Functions
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►In each case when , the quantity multiplying supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0.
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19.10.2
16: 19.16 Definitions
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►Just as the elementary function (§19.2(iv)) is the degenerate case
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►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function
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§19.16(iii) Various Cases of
… ►The only cases that are integrals of the third kind are those in which at least one is a positive integer. All other elliptic cases are integrals of the second kind. …17: 13.24 Series
18: 14.20 Conical (or Mehler) Functions
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14.20.3
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►Special cases:
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14.20.13
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14.20.14
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►For the case of purely imaginary order and argument see Dunster (2013).
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19: 22.10 Maclaurin Series
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22.10.7
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22.10.8
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22.10.9
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►The radius of convergence is the distance to the origin from the nearest pole in the complex -plane in the case of (22.10.4)–(22.10.6), or complex -plane in the case of (22.10.7)–(22.10.9); see §22.17.
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20: 19.24 Inequalities
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►Inequalities for are included as the case
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►For , , and , the complete cases of and satisfy
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►Inequalities for and are included as special cases (see (19.16.6) and (19.16.5)).
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►Special cases with are (19.24.8) (because of (19.16.20), (19.16.23)), and
…The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases.
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